L(s) = 1 | + 5·5-s + 35.5·7-s − 44.7·11-s + 77.8·13-s − 120.·17-s + 121.·19-s + 152.·23-s + 25·25-s + 187.·29-s − 161.·31-s + 177.·35-s + 13.3·37-s + 188.·41-s − 81.5·43-s + 48.6·47-s + 923.·49-s − 707.·53-s − 223.·55-s + 16.4·59-s − 743.·61-s + 389.·65-s − 59.2·67-s + 144.·71-s + 657.·73-s − 1.59e3·77-s − 454.·79-s − 165.·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.92·7-s − 1.22·11-s + 1.66·13-s − 1.71·17-s + 1.46·19-s + 1.38·23-s + 0.200·25-s + 1.20·29-s − 0.934·31-s + 0.859·35-s + 0.0592·37-s + 0.716·41-s − 0.289·43-s + 0.150·47-s + 2.69·49-s − 1.83·53-s − 0.548·55-s + 0.0363·59-s − 1.56·61-s + 0.742·65-s − 0.108·67-s + 0.241·71-s + 1.05·73-s − 2.35·77-s − 0.646·79-s − 0.218·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.143023640\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.143023640\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
good | 7 | \( 1 - 35.5T + 343T^{2} \) |
| 11 | \( 1 + 44.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 77.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 120.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 121.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 152.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 161.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 13.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 188.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 81.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 48.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 707.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 16.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 743.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 59.2T + 3.00e5T^{2} \) |
| 71 | \( 1 - 144.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 657.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 454.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 165.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 535.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 436.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.308065400188016437950983007626, −8.606143107303739713330374980298, −7.971411654190434686163805628343, −7.10865620868416727403822924859, −5.96317333640908672804300542581, −5.08311201516986643901376457687, −4.53577695956556733897742700853, −3.07567892528818894291174251415, −1.92210610733812861614694919768, −1.00768666447779645305772082435,
1.00768666447779645305772082435, 1.92210610733812861614694919768, 3.07567892528818894291174251415, 4.53577695956556733897742700853, 5.08311201516986643901376457687, 5.96317333640908672804300542581, 7.10865620868416727403822924859, 7.971411654190434686163805628343, 8.606143107303739713330374980298, 9.308065400188016437950983007626